A Novel Application of Expansive Mappings and Cone Theory for Third-Order Impulsive Dynamic Equations

We establish novel existence criteria for solutions to a class of third-order impulsive nonlinear dynamic equations on time scales, subject to homogeneous boundary conditions. Although first and second-order impulsive dynamic equations are well studied, the analysis of third-order problems remains comparatively underdeveloped. To address this gap, we employ very recent advancements in fixed point theory, utilizing innovative concepts involving expansive mappings and operator addition. This approach enables us to first prove the existence of at least one solution and subsequently demonstrate the presence of at least two distinct solutions. A significant advantage of our methodology is that it does not impose restrictions on boundedness on the nonlinear components of the equation, a requirement common in most existing studies. The applicability and practical relevance of our theoretical results are validated through an illustrative example.